具变号权函数的二阶微分系统正解的存在性
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摘要
将二阶微分方程边值问题推广到n维二阶微分系统中,并研究n维二阶微分系统在权函数变号的情况下正解的存在性。首先,将二阶微分系统转化成与原微分系统等价的积分系统;其次,根据得到的积分系统的具体表达式以及与其对应的格林函数的性质,构造适当的范数、锥和积分算子;最后,运用范数形式的锥拉伸与锥压缩不动点定理,结合微分系统中权函数变号的特点,对非线性项构造适当的条件,使其满足不动点定理,得到积分算子不动点的存在性,进而得到原微分系统正解的存在性。运用不动点定理,得到积分算子至少存在一个不动点,进而得到原二阶微分系统至少存在一个正解。原具变号权函数的二阶微分系统至少存在一个正解。
We extend the boundary value problem of second-order differential equation to ndimensional second-order differential systems,and consider the existence of positive solutions for ndimensional second-order differential system with indefinite weight function.Firstly,we transform the second-order differential system into integral system which equal to original differential system.Secondly,we construct suitable norm,cone and integral operator,according to the specific form of integral system and the properties of Green's function.Finally,using the fixed-point theorem of cone expansion and compression of norm type,and combining the characteristics of the indefinite weight function,we construct appropriate conditions of the nonlinear term,such that the fixedpoint theorem is satisfied.And we obtain the existence of the fixed point of the operator,and then the existence of the positive solutions of original differential system is obtained.Using the fixed point theorem,we obtain that there is at least one fixed point of the integral operator,and then there is at least one positive solution to original second-order differential system.There is at least one positive solution for original second-order differential system with indefinite weight function.
We extend the boundary value problem of second-order differential equation to ndimensional second-order differential systems,and consider the existence of positive solutions for ndimensional second-order differential system with indefinite weight function.Firstly,we transform the second-order differential system into integral system which equal to original differential system.Secondly,we construct suitable norm,cone and integral operator,according to the specific form of integral system and the properties of Green's function.Finally,using the fixed-point theorem of cone expansion and compression of norm type,and combining the characteristics of the indefinite weight function,we construct appropriate conditions of the nonlinear term,such that the fixedpoint theorem is satisfied.And we obtain the existence of the fixed point of the operator,and then the existence of the positive solutions of original differential system is obtained.Using the fixed point theorem,we obtain that there is at least one fixed point of the integral operator,and then there is at least one positive solution to original second-order differential system.There is at least one positive solution for original second-order differential system with indefinite weight function.
引文
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[4]MA Ruyun,HAN Xiaoling.Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function[J].Appl Math Comput,2009,215(3):1077-1083.
[5]GAO Chenghua,DAIGuowei,MA Ruyun.Existence of positive solutions to discrete second-order boundary value problems with indefinite weight[J].Adv Differ Equ,2012,2012(1):1-10.
[6]WANG Fanglei,AN Yukun.Existence of doubly periodic solutions for a class of telegraph system with indefinite weight[J].Bull Sci Math,2013,137(8):1007-1017.
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[13]FENG Meiqiang,ZHANG Xuemei,GE Weigao.Existence theorems for a second order nonlinear differential equation with nonlocal boundary conditions and their applications[J].J Appl Math Comput,2010,33(1/2):137-153.
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[15]卢高丽,冯美强.具偏差变元的二阶微分方程多个正解及其应用[J].中国科技论文,2016,11(5):571-575.
[16]WANG Fanglei,AN Yukun.Positive solutions for a second-order differential system[J].J Math Anal Appl,2011,373(2):370-375.
[17]曹君艳,王全义.一类二阶微分方程两点边值问题的正解存在性[J].华侨大学学报(自然科学版),2010,31(1):113-117.
[18]郭大钧.非线性泛函分析[M].3版.北京:高等教育出版社,2015.
[19]冯美强,张学梅.非线性常微分方程边值问题应用研究[M].北京:北京科学技术出版社,2010.
[2]JIANG Xiufen,YAO Qingliu.An existence theorem of positive solution for a semilinear two-point BVP with change of sign[J].Math Appl,2001,14(3):68-71.
[3]WANG Fanglei,AN Yukun.On positive solutions for a second order differential system with indefinite weight[J].Elsevier Science Inc,2015,259:753-761.
[4]MA Ruyun,HAN Xiaoling.Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function[J].Appl Math Comput,2009,215(3):1077-1083.
[5]GAO Chenghua,DAIGuowei,MA Ruyun.Existence of positive solutions to discrete second-order boundary value problems with indefinite weight[J].Adv Differ Equ,2012,2012(1):1-10.
[6]WANG Fanglei,AN Yukun.Existence of doubly periodic solutions for a class of telegraph system with indefinite weight[J].Bull Sci Math,2013,137(8):1007-1017.
[7]ALBERTO B,FABIIO Z.Second-order ordinary differential equations with indefinite weight:the Neumann boundary value problem[J].Ann Mat Pur Appl,2015,194(2):451-478.
[8]ALBERTO B,MAURIZIO G.Multiple solutions to Neumann problems with indefinite weight and bounded nonlinearities[J].J Dyn Differ Equ,2016,28(1):1-21.
[9]RAFAEL B,HAIMB,ELLIOTT H L.The thomas-fermi-von weizsacker theory of atoms and molecules[J].Commun Math Phys,1981,79(2):167-180.
[10]ELLIOTT H L.Thomas-Fermi and related theories of atoms and molecules[J].Rev Mod Phys,1981,53:603-641.
[11]WILLIAM E K,PAUL L C.On the stability of nonlinear space-dependent reactor kinetics[J].Nucl Sci Eng,1968,31(1):67-79.
[12]ZHANG Xuemei,FENG Meiqiang.Positive solutions for a second-order differential equation with integral boundary conditions and deviating arguments[J].Bound Value Probl,2015,2015(1):1-21.
[13]FENG Meiqiang,ZHANG Xuemei,GE Weigao.Existence theorems for a second order nonlinear differential equation with nonlocal boundary conditions and their applications[J].J Appl Math Comput,2010,33(1/2):137-153.
[14]CHEN Haibo,LI Peiluan.Three-point boundary value problems for second-order ordinary differential equations in Banach spaces[J].Comput Math Appl,2008,56(7):1852-1860.
[15]卢高丽,冯美强.具偏差变元的二阶微分方程多个正解及其应用[J].中国科技论文,2016,11(5):571-575.
[16]WANG Fanglei,AN Yukun.Positive solutions for a second-order differential system[J].J Math Anal Appl,2011,373(2):370-375.
[17]曹君艳,王全义.一类二阶微分方程两点边值问题的正解存在性[J].华侨大学学报(自然科学版),2010,31(1):113-117.
[18]郭大钧.非线性泛函分析[M].3版.北京:高等教育出版社,2015.
[19]冯美强,张学梅.非线性常微分方程边值问题应用研究[M].北京:北京科学技术出版社,2010.